Assume that \(\mathcal {M}\)  is compact. Let \(\eta (u,v) = \theta \)  be the transformation to the tubular neighborhood (see, for instance, [1]}), and assume that the Hessian of the constraint function \(g\)  and \(\gamma _n\)  are equicontinuous at the point \(v = 0\)  (that is, the families \(\gamma _{u,n}(v) := \gamma _n(\eta (u,v))\)  and \(\frac{\partial ^2 g_u (v)}{\partial v^2} := \frac{\partial ^2 g (\eta (u,v))}{\partial v^2}\)  are equicontinuous at \(v = 0\)  over \(u,n\) ). Then, for \(H(\theta ) = \left( \det \left| \nabla _\theta g (\nabla _\theta g)^T \right| \right)^{-1/2},\)
\(\int _{\mathcal {M} } \left| \frac{ r_n(\theta | \mathbf {y}) H(\theta ) }{\int _{\mathcal {M} }r_n(\theta | y) H(\theta ) d \lambda (\theta ) } - \frac{ \phi _{\theta _0, nI(\theta _0)}(\theta ) H(\theta ) }{\int _{\mathcal {M} } \phi _{\theta _0, nI(\theta _0)}(\theta )H(\theta ) d \lambda (\theta ) }\right| d \lambda (\theta ) \overset{P_{\theta _0}}{\rightarrow } 0.\) 

